Cosmological aspects of the Eisenhart–Duval lift

Cariglia, Marco; Galajinsky, Anton; Gibbons, G. W.; Horváthy, P. A.

doi:10.1140/epjc/s10052-018-5789-x

Cited by 47 publications

(32 citation statements)

References 68 publications

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“…In consequence the Niederer transformation can be useful to solve some geodesic equations for plane gravitational waves (or other problems where time dependent linear oscillators occur). Moreover, such properties of the Niederer transformation have a reflection in a geometric picture obtained by means of the Eisenhart-Duval lift [23,65] (see also [66,67] and references therein). Namely, extending the Niederer map by adding the following transformation rule…”

Section: Niederer's Map and Lorentz's Force Equationmentioning

confidence: 97%

From polarized gravitational waves to analytically solvable electromagnetic beams

Andrzejewski

Prencel

2019

Phys. Rev. D

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Using the correspondence between solutions of gravitational and gauge theories (the so-called classical double copy conjecture) some electromagnetic fields with vortices are constructed, for which the Lorentz force equations are analytically solvable. The starting point is a certain class of plane gravitational waves exhibiting the conformal symmetry. The notion of the Niederer transformation, crucial for the solvability, is analysed in the case of the Lorentz force equation on the curved spacetimes as well as its derivation by means of integrals of motion (associated with conformal generators preserving these vortices) is presented. Furthermore, some models discussed recently in the context of the intense laser beams are constructed from their gravitational counterparts, with the special emphasis put on the focusing property, and new solvable examples are presented.

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Section: Niederer's Map and Lorentz's Force Equationmentioning

confidence: 97%

From polarized gravitational waves to analytically solvable electromagnetic beams

Andrzejewski

Prencel

2019

Phys. Rev. D

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“…Computing the Weyl tensor, one can verify that its vanishing requires ∆ log(ϕ) = 0, which in turn implies that the metric (4a) is flat. This makes the cosmological applications in the spirit of a recent work [27] problematic. Furthermore, it is worth recalling that reductions of the Goryachev-Chaplygin and Kovalevskaya tops, which are obtained by discarding a cyclic variable, result in 2D integrable systems in curved space possessing cubic and quartic integrals of motion, respectively.…”

Section: Resultsmentioning

confidence: 99%

Eisenhart lift of 2-dimensional mechanics

Fordy

Galajinsky

2019

Eur. Phys. J. C

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The Eisenhart lift is a variant of geometrization of classical mechanics with d degrees of freedom in which the equations of motion are embedded into the geodesic equations of a Brinkmann-type metric defined on (d+2)-dimensional spacetime of Lorentzian signature. In this work, the Eisenhart lift of 2-dimensional mechanics on curved background is studied. The corresponding 4-dimensional metric is governed by two scalar functions which are just the conformal factor and the potential of the original dynamical system. We derive a conformal symmetry and a corresponding quadratic integral, associated with the Eisenhart lift. The energy-momentum tensor is constructed which, along with the metric, provides a solution to the Einstein equations. Uplifts of 2-dimensional superintegrable models are discussed with a particular emphasis on the issue of hidden symmetries. It is shown that for the 2-dimensional Darboux-Koenigs metrics, only type I can result in Eisenhart lifts which satisfy the weak energy condition. However, some physically viable metrics with hidden symmetries are presented.

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“…It would be very interesting to generalize the system with harmonic trap that we considered for the case of D(2, 1; α) superconformal mechanics and to look for its relation with the systems from [72] in the light of the conformal bridge transformation. In another but somehow related direction, it could be interesting to study this system and its hidden symmetries from the perspective of Eisenhart-Duval lift [73] and Killing-Yano tensors [1].…”

Section: Discussion and Outlookmentioning

confidence: 99%

Hidden symmetry and (super)conformal mechanics in a monopole background

Inzunza

Plyushchay

Wipf

2020

J. High Energ. Phys.

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We study classical and quantum hidden symmetries of a particle with electric charge e in the background of a Dirac monopole of magnetic charge g subjected to an additional central potential V (r) = U (r) + (eg) 2 /2mr 2 with U (r) = 1 2 mω 2 r 2 , similar to that in the one-dimensional conformal mechanics model of de Alfaro, Fubini and Furlan (AFF). By means of a non-unitary conformal bridge transformation, we establish a relation of the quantum states and of all symmetries of the system with those of the system without harmonic trap, U (r) = 0. Introducing spin degrees of freedom via a very special spin-orbit coupling, we construct the osp(2|2) superconformal extension of the system with unbroken N = 2 Poincaré supersymmetry and show that two different superconformal extensions of the one-dimensional AFF model with unbroken and spontaneously broken supersymmetry have a common origin. We also show a universal relationship between the dynamics of a Euclidean particle in an arbitrary central potential U (r) and the dynamics of a charged particle in a monopole background subjected to the potential V (r).To solve the vector equation in (2.4) in the case J 2 > ν 2 , which we will assume in what follows, we decompose n into the component parallel to the angular momentum and the orthogonal component,whereĴ is the unit vector in the direction of J . Since the parallel component n is constant, we conclude that the orthogonal component n ⊥ (t) has constant length and thus describes a circle in the plane orthogonal to J , n ⊥ (t) = n ⊥ (0) cos ϕ(t) +Ĵ × n ⊥ (0) sin ϕ(t) .(2.15) Here, θ, ϕ and ψ are the Euler angles, 0 ≤ ϕ, ψ < 2π, 0 ≤ θ < π, and J i J i = K a K a = J 2 . The common eigenstates of J 2 , J 3 and K 3 satisfying relations

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Cosmological aspects of the Eisenhart–Duval lift

Cited by 47 publications

References 68 publications

From polarized gravitational waves to analytically solvable electromagnetic beams

From polarized gravitational waves to analytically solvable electromagnetic beams

Eisenhart lift of 2-dimensional mechanics

Hidden symmetry and (super)conformal mechanics in a monopole background

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